Dynamics of Boolean Networks

Networks are used to model systems consisting of many interacting units. The topology of networks has been studied extensively while there are still many open questions concerning the dynamics of and on networks. Boolean networks refer to a class of dynamics on networks; it is the simplest possible dynamics which allows for analytical studies and easy computer implementation. Applications of networks in general and Boolean networks in particular can be found in numerous fields, ranging from chemistry, biology, economy, computer sciences, linguistics, to sociology and geology. A classical Boolean network was introduced by Stuart Kauffman as a simple model for gene regulation. The Boolean state of a node is determined by a Boolean function whose arguments are the states of its randomly chosen inputs. The inputs are those nodes from which a edge of the network leads to the node under consideration. Key quantities for the dynamics are the number and size of attractors. An attractor is a recurrent set of a network’s states. Although the classical model of random Boolean networks with synchronous updating has been introduced in 1969 it took 30 years for an analytical understanding of its key quantities. The present work studies variations of the classical implementation of a random Boolean network, in particular models with differently defined dynamics (non-synchronous updating, the ensemble of threshold functions instead of all Boolean functions) and systems with other connection patterns (scale-free in-degree distribution). To study the characteristics of the dynamics of those variations both, computer simulations and analytic methods, are used. For the classical critical Boolean network (with synchronous updating) it is known that both, the mean number and the length of attractors, diverge faster than any power law with the number of nodes. It is shown how this changes for asynchronous updating schemes, in particular for a deterministic asynchronous one. Boolean threshold functions lead new phases of the dynamics which are not predicted by the usual mean-field considerations, real genetic networks might therefore also have a richer dynamical behaviour than the classical dynamical phases. For the scale-free topology, the number of the non-frozen nodes scales in a different way as in networks with a fixed number of inputs. Here, it is possible to analytically explain numerical findings in literature.